Abstract
A trigonometrically fitted diagonally implicit two-derivative Runge–Kutta method (TFDITDRK) is used for the numerical integration of first-order delay differential equations (DDEs) which possesses oscillatory solutions. Using the trigonometrically fitted property, a three-stage fifth-order diagonally implicit two- derivative Runge–Kutta (DITDRK) method is derived. Here, we employed trigonometric interpolation for the approximation of the delay term. The curves of efficiency based on the log of maximum errors against the log of function evaluations and the CPU time spent to perform the integration are plotted, which then clearly illustrated the superiority of the trigonometrically fitted DITDRK method in comparison with its original method and other existing diagonally implicit Runge–Kutta (DIRK) methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ababneh O, Ahmad R, Ismail E (2009) Design of new diagonally implicit Runge–Kutta methods for stiff problems. Appl Math Sci 3:2241–2253
Ahmad NA, Senu N, Ismail F (2016) Phase-fitted and amplification-fitted diagonally implicit Runge–Kutta method for the numerical solution of periodic problems. Asian J Math Comput Res 12:252–264
Ahmad NA, Senu N, Ibrahim ZB, Othman M (2019a) Trigonometrically-fitted diagonally implicit two derivative Runge–Kutta method for the numerical solution of periodical IVPs. ASM Sci J 12:50–59
Ahmad NA, Senu N, Ibrahim ZB, Othman M (2019b) Diagonally implicit two derivative Runge–Kutta methods for solving first order initial value problems. J Abstr Comput Math 1:18–36
Baker CT, Paul CA (1994) Computing stability regions-Runge–Kutta methods for delay differential equations. IMA J Numer Anal 14:347–362
Bartoszewski Z, Jackiewicz Z (2002) Stability analysis of two-step Runge–Kutta methods for delay differential equations. Comput Math Appl 44:83–93
Bocharov G, Marchuk G, Romanyukha A (1996) Numerical solution by LMMS of stiff delay differential systems modelling an immune response. Numer Math 73:131–148
Conte SD, De Boor C (2017) Elementary numerical analysis: an algorithmic approach. SIAM
In’t Hout K (1992) A new interpolation procedure for adapting Runge–Kutta methods to delay differential equations. BIT Numer Math 32:634–649
Ishak F, Suleiman M, Omar Z (2008) Two-point predictor–corrector block method for solving delay differential equations. MATEMATIKA: Malaysian J Indust Appl Math 24:131–140
Ishak F, Majid ZA, Suleiman M (2010) Two-point block method in variable stepsize technique for solving delay differential equations. J Mater Sci Eng 4:86–90
Ismail F, Suleiman M (2000) The P-stability and Q-stability of singly diagonally implicit Runge–Kutta method for delay differential equations. Int J Comput Math 76:267–277
Ismail F, Al-Khasawneh RA, Suleiman M (2002) Numerical treatment of delay differential equations by Runge–Kutta method using hermite interpolation. Matematika 18:79–90
Karoui A (1992) On the numerical solution of delay differential equations. University of Ottawa
Maset S (2000) Stability of Runge–Kutta methods for linear delay differential equations. Numer Math 87:355–371
Mechee M, Ismail F, Senu N, Siri Z (2013) Directly solving special second order delay differential equations using Runge–Kutta–Nyström method. Math Probl Eng 295:7
Morrison TM, Rand RH (2007) 2:1 resonance in the delayed nonlinear Mathieu equation. Nonlinear Dyn 50:341–352
Neves KW (1981) Control of interpolatory error in retarded differential equations. ACM Trans Math Softw (TOMS) 7:421–444
Oberle HJ, Pesch HJ (1981) Numerical treatment of delay differential equations by hermite interpolation. Numer Math 37:235–255
Rihan FA (2013) Delay differential equations in biosciences: parameter estimation and sensitivity analysis. In: Recent advances in applied Mathemat290 ICS and computational methods: proceedings of the 2013 international conference on applied mathematics and computational methods, Venice, Italy, September 2013, pp 50–58
Schmitt K (1971) Comparison theorems for second order delay differential equations. Rocky Mt J Math 1:459–467
Shampine LF, Thompson S (2001) Solving DDEs in Matlab. Appl Numer Math 37:441–458
Suleiman M (2012) Two and three point one-step block methods for solving delay differential equations. J Qual Meas Anal JQMA 8(1):29–41
Taiwo O, Odetunde O (2010) On the numerical approximation of delay differential equations by a decomposition method. Asian J Math Stat 3:237–243
Yann Seong H, Abdul Majid Z, Ismail F (2013) Solving second-order delay differential equations by direct Adams-Moulton method. Math Probl Eng 7
Zennaro M (1986) P-stability properties of Runge–Kutta methods for delay differential equations. Numer Math 49:305–318
Acknowledgements
The authors acknowledge the financial support from Universiti Putra Malaysia under Putra-IPB Grant: GP-IPB/2017/954240
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Valeria Neves Domingos Cavalcanti.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Senu, N., Ahmad, N.A., Othman, M. et al. Numerical study for periodical delay differential equations using Runge–Kutta with trigonometric interpolation. Comp. Appl. Math. 41, 25 (2022). https://doi.org/10.1007/s40314-021-01728-8
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-021-01728-8
Keywords
- Diagonally implicit two-derivative Runge–Kutta method
- Delay differential equations
- Initial value problems
- Trigonometrically fitted
- Trigonometric interpolation